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Lecture 14  3/19/2010
Discrete Stochastic Processes
1
DISCRETE
STOCHASTIC
PROCESSES
Lecture 14
Section 3.6
Little’s Theorem
Average waiting time of M/G/1 queue  the PK Formula
“Pasta Property”  Poisson Arrivals See Time Averages
Section 3.7
Mention “Delayed Renewal Processes”
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Discrete Stochastic Processes
2
SteadyState Renewal & Reward Theory
LongTerm Time Averages
Renewals
Strong Law for Renewals
()
1
lim
1
t
Nt
wp
tX
→∞
=
____
____
____
____
____ ____ ____ ____ ____ ____
____
____
____
Rewards
Strong Law for Renewal Rewards
{ }
0
1
lim
t
n
t
E R
Rd
ττ
→∞
=
∫
SteadyState Expectations
Elementary Renewal Theorem
( )
{ }
( )
1
lim
lim
tt
ENt
mt
X
→∞
→∞
=
=
Blackwell’s Theorem
( ) ( )
{ }
(
)(
)
1
lim
lim
X
δ
→∞
→∞
+−
=
=
Arithmetic Case:
( ) ( )
1
lim
t
mt nd mt
nd
X
→∞
=
Key Renewal Theorem Corollary 1
{}
{ }
lim
n
t
E R
ERt
X
→∞
=
Arithmetic Case:
{ }
lim
n
k
E R
ERk
d
X
→∞
=
SteadyState Probabilities
From Blackwell’s Theorem
( ) ( )
{ }
lim
1
t
PNt
→∞
+
−=
=
( ]
{ }
lim
exactly 1 renewal in
,
t
Pt
t
→∞
+
=
o
X
−
Arithmetic Case:
lim
exactly 1 renewal at
t
d
Pn
d
X
→∞
=
SteadyState Densities
Assume X has a density
() ()
( )
,
lim
,
, 0
X
t
Zt Xt
fx
f zx
z x
X
→∞
=
≤<
( )
1
lim
lim
X
Yt
Zt
Fz
fz
f z
X
→∞
→∞
−
==
( )
lim
X
t
Xt
x
X
→∞
=
Lecture 14  3/19/2010
Discrete Stochastic Processes
3
()
Lt
=
0
1
[0, ]
t
L
d
t
ττ
=
∫
[0, ]
[0, ]
Wt
A t
t
Little's Theorem
A(t)
D(t)
0S
S
12
W
1
3
W
A(t)= Arrival process
D(t)=Departure process
W
2
Renewal epochs
number of customers in system (service + queue) at time
t
is a reward
function = A(t) – D(t)
Time averaged number of customers over [0,t]
=
n
W
=
wait (in queue and service) for nth customer. (Neither independent nor
identically distributed! Why?)
Average wait for all customers arriving in [0,t]
=
For
t
in empty period before S
1
:
i.e., (time averaged occupancy) = (average wait per customer)
∗
(average arrival rate)
=
*
( )
( )
11
123
0
(
)
[0, ]
(
)
At
kk
t
WW
A t
L
d
W
tt
t
A
t
t
==
+
+
=
=
∑∑
∫
( )
1
[0, ]
k
k
W
A t
=
=
∑
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Discrete Stochastic Processes
4
λ
W
∗
..
1
E[L ]
(
)
n
wp
L
X
=
This is also true for t in any empty period before S
n
,
n = 1, 2, 3
, 
Little’s theorem asserts that this works in the limit of large t:
Little’s Theorem (for G/G/1 Queue)
=
L
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This note was uploaded on 10/21/2010 for the course EE 5581 taught by Professor Moon,j during the Spring '08 term at Minnesota.
 Spring '08
 Moon,J

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